Calculus Examples

$\int 2 \sqrt{x} \sqrt{1 + {(\frac{1}{\sqrt{x}})}^{2}} d x$

Step 1

Simplify.

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Step 1.1

Combine using the product rule for radicals.

$\int 2 \sqrt{(1 + {(\frac{1}{\sqrt{x}})}^{2}) x} d x$

Step 1.2

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\int 2 \sqrt{(1 + {(\frac{1}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}})}^{2}) x} d x$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\int 2 \sqrt{(1 + {(\frac{\sqrt{x}}{\sqrt{x} \sqrt{x}})}^{2}) x} d x$

Step 1.3.2

Raise $\sqrt{x}$ to the power of $1$ .

$\int 2 \sqrt{(1 + {(\frac{\sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}})}^{2}) x} d x$

Step 1.3.3

Raise $\sqrt{x}$ to the power of $1$ .

$\int 2 \sqrt{(1 + {(\frac{\sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}})}^{2}) x} d x$

Step 1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 2 \sqrt{(1 + {(\frac{\sqrt{x}}{{\sqrt{x}}^{1 + 1}})}^{2}) x} d x$

Step 1.3.5

Add $1$ and $1$ .

$\int 2 \sqrt{(1 + {(\frac{\sqrt{x}}{{\sqrt{x}}^{2}})}^{2}) x} d x$

Step 1.3.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int 2 \sqrt{(1 + {(\frac{\sqrt{x}}{{(x^{\frac{1}{2}})}^{2}})}^{2}) x} d x$

Step 1.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int 2 \sqrt{(1 + {(\frac{\sqrt{x}}{x^{\frac{1}{2} \cdot 2}})}^{2}) x} d x$

Step 1.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\int 2 \sqrt{(1 + {(\frac{\sqrt{x}}{x^{\frac{2}{2}}})}^{2}) x} d x$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\int 2 \sqrt{(1 + {(\frac{\sqrt{x}}{x^{\frac{2}{2}}})}^{2}) x} d x$

Step 1.3.6.4.2

Rewrite the expression.

$\int 2 \sqrt{(1 + {(\frac{\sqrt{x}}{x^{1}})}^{2}) x} d x$

Step 1.3.6.5

Simplify.

$\int 2 \sqrt{(1 + {(\frac{\sqrt{x}}{x})}^{2}) x} d x$

Step 1.4

Apply the product rule to $\frac{\sqrt{x}}{x}$ .

$\int 2 \sqrt{(1 + \frac{{\sqrt{x}}^{2}}{x^{2}}) x} d x$

Step 1.5

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int 2 \sqrt{(1 + \frac{{(x^{\frac{1}{2}})}^{2}}{x^{2}}) x} d x$

Step 1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int 2 \sqrt{(1 + \frac{x^{\frac{1}{2} \cdot 2}}{x^{2}}) x} d x$

Step 1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\int 2 \sqrt{(1 + \frac{x^{\frac{2}{2}}}{x^{2}}) x} d x$

Step 1.5.4

Cancel the common factor of $2$ .

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Step 1.5.4.1

Cancel the common factor.

$\int 2 \sqrt{(1 + \frac{x^{\frac{2}{2}}}{x^{2}}) x} d x$

Step 1.5.4.2

Rewrite the expression.

$\int 2 \sqrt{(1 + \frac{x^{1}}{x^{2}}) x} d x$

Step 1.5.5

Simplify.

$\int 2 \sqrt{(1 + \frac{x}{x^{2}}) x} d x$

Step 1.6

Cancel the common factor of $x$ and $x^{2}$ .

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Step 1.6.1

Raise $x$ to the power of $1$ .

$\int 2 \sqrt{(1 + \frac{x^{1}}{x^{2}}) x} d x$

Step 1.6.2

Factor $x$ out of $x^{1}$ .

$\int 2 \sqrt{(1 + \frac{x \cdot 1}{x^{2}}) x} d x$

Step 1.6.3

Cancel the common factors.

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Step 1.6.3.1

Factor $x$ out of $x^{2}$ .

$\int 2 \sqrt{(1 + \frac{x \cdot 1}{x \cdot x}) x} d x$

Step 1.6.3.2

Cancel the common factor.

$\int 2 \sqrt{(1 + \frac{x \cdot 1}{x \cdot x}) x} d x$

Step 1.6.3.3

Rewrite the expression.

$\int 2 \sqrt{(1 + \frac{1}{x}) x} d x$

Step 1.7

Write $1$ as a fraction with a common denominator.

$\int 2 \sqrt{(\frac{x}{x} + \frac{1}{x}) x} d x$

Step 1.8

Combine the numerators over the common denominator.

$\int 2 \sqrt{\frac{x + 1}{x} x} d x$

Step 1.9

Combine $\frac{x + 1}{x}$ and $x$ .

$\int 2 \sqrt{\frac{(x + 1) x}{x}} d x$

Step 1.10

Cancel the common factor of $x$ .

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Step 1.10.1

Cancel the common factor.

$\int 2 \sqrt{\frac{(x + 1) x}{x}} d x$

Step 1.10.2

Divide $x + 1$ by $1$ .

$\int 2 \sqrt{x + 1} d x$

Step 2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \sqrt{x + 1} d x$

Step 3

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $x + 1$ .

$\frac{d}{d x} [x + 1]$

Step 3.1.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [1]$

Step 3.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0$

Step 3.1.5

Add $1$ and $0$ .

$1$

Step 3.2

Rewrite the problem using $u$ and $d u$ .

$2 \int \sqrt{u} d u$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$2 \int u^{\frac{1}{2}} d u$

Step 5

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

$2 (\frac{2}{3} u^{\frac{3}{2}} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $2 (\frac{2}{3} u^{\frac{3}{2}} + C)$ as $2 (\frac{2}{3}) u^{\frac{3}{2}} + C$ .

$2 (\frac{2}{3}) u^{\frac{3}{2}} + C$

Step 6.2

Simplify.

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Step 6.2.1

Combine $2$ and $\frac{2}{3}$ .

$\frac{2 \cdot 2}{3} u^{\frac{3}{2}} + C$

Step 6.2.2

Multiply $2$ by $2$ .

$\frac{4}{3} u^{\frac{3}{2}} + C$

Step 7

Replace all occurrences of $u$ with $x + 1$ .

$\frac{4}{3} {(x + 1)}^{\frac{3}{2}} + C$